Hajimemashite - "Nice to meet you." used when meeting someone for the first time after introductions.
Dozoyoroshiku - "Please be kind to me" or literally translated "Please give favors to me". used when meeting someone for the first time that you will be working or staying with for a while, usually said after Hajimemashite.
Onegaishimasu - "Could you please" said after asking someone to do something for you